ment relating to our perceiving nature.[50]

[50] The same criticism can be urged against Kant's appeal to
the necessity of _constructing_ geometrical figures. The
conclusion drawn from the necessity of construction is stated
thus: "If the object (the triangle) were something in itself
without relation to you the subject, how could you say that
that which lies necessarily in your subjective conditions of
constructing a triangle must also necessarily belong to the
triangle in itself?" (B. 65, M. 39). Kant's thought is that
the laws of the mind's constructing nature must apply to
objects, if, and only if, the objects are the mind's own
construction. Hence it is open to the above criticism if, in
the criticism, 'construct' be substituted for 'perceive'.

This difficulty is concealed from Kant by his insistence on the
_perception_ of space involved in geometrical judgements. This leads
him at times to identify the judgement and the perception, and,
therefore, to speak of the judgement as a perception. Thus we find him
saying that mathematical judgements are always _perceptive_,[51] and
that "It is only possible for my perception to precede the actuality
of the object and take place as _a priori_ knowledge, if &c."[52]
Hence, if, in addition, a geometrical judgement, as being a judgement
about a necessity, be identified with a necessity of judging, the
conformity of things to these universal judgements will become the
conformity of things to rules or necessities of our judging, i. e. of
our perceiving nature, and Kant's conclusion will at once follow.[53]
Unfortunately for Kant, a geometrical judgement, however closely
related to a perception, must itself, as the apprehension of what is
necessary and universal, be an act of thought rather than of
perception, and therefore the original problem of the conformity of
things to our mind can be forced upon him again, even after he thinks
that he has solved it, in the new form of that of the conformity
within the mind of perceiving to thinking.

[51] _Prol._, Sec. 7.

[52] _Prol._, Sec. 9.

[53] Cf. (_Introduction_, B. xvii, M. xxix): "But if the
object (as object of the senses) conforms to the nature of
our faculty of perception, I can quite well represent to
myself the possibility of _a priori_ knowledge of it [i. e.
mathematical knowledge]."

The fact is simply that the universal validity of geom